<abstract xmlns="http://www.w3.org/1999/xhtml">

<p>In this paper we study the density in the real line of oscillating sequences of the form
<disp-formula>
<alternatives>
<graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_udt-2022-0003_eq_001.png"></graphic>
<math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mrow><msub><mrow><mrow><mrow><mo>(</mo><mrow><mi>g</mi><mrow><mo>(</mo><mi>k</mi><mo>)</mo></mrow><mo>⋅</mo><mi>F</mi><mrow><mo>(</mo><mrow><mi>k</mi><mi>α</mi></mrow><mo>)</mo></mrow></mrow><mo>)</mo></mrow></mrow></mrow><mrow><mi>k</mi><mo>∈</mo><mi>ℕ</mi></mrow></msub><mo>,</mo></mrow></math>
<tex-math>{\left( {g\left( k \right) \cdot F\left( {k\alpha } \right)} \right)_{k \in \mathbb{N}}},</tex-math>
</alternatives>
</disp-formula>
where <italic>g</italic> is a positive increasing function and <italic>F</italic> a real continuous 1-periodic function. This extends work by Berend, Boshernitzan and Kolesnik [Distribution Modulo 1 of Some Oscillating Sequences I-III] who established differential properties on the function <italic>F</italic> ensuring that the oscillating sequence is dense modulo 1.</p>
<p>More precisely, when <italic>F</italic> has finitely many roots in [0, 1), we provide necessary and also sufficient conditions for the oscillating sequence under consideration to be dense in ℝ. All the results are stated in terms of the Diophantine properties of <italic>α</italic>, with the help of the theory of continued fractions.</p>
</abstract>

In this paper we study the density in the real line of oscillating sequences of the form



(g(k)⋅F(kα))k∈ℕ,
{\left( {g\left( k \right) \cdot F\left( {k\alpha } \right)} \right)_{k \in \mathbb{N}}},


where g is a positive increasing function and F a real continuous 1-periodic function. This extends work by Berend, Boshernitzan and Kolesnik [Distribution Modulo 1 of Some Oscillating Sequences I-III] who established differential properties on the function F ensuring that the oscillating sequence is dense modulo 1.
More precisely, when F has finitely many roots in [0, 1), we provide necessary and also sufficient conditions for the oscillating sequence under consideration to be dense in ℝ. All the results are stated in terms of the Diophantine properties of α, with the help of the theory of continued fractions.

Density of Oscillating Sequences in the Real Line

ioannis.tsokanos@postgrad.manchester.ac.uk

Department of Mathematics, Faculty of Science and Engineering

Uniform distribution theory

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

{"article-title":"Density of Oscillating Sequences in the Real Line"}

In this paper we study the density in the real line of oscillating sequences of the form



(g(k)⋅F(kα))k∈ℕ,
{\left( {g\left( k \right) \cdot F\left( {k\alpha...

Density of Oscillating Sequences in the Real Line

Published Online: May 31, 2022

Page range: 105 - 130

Received: May 21, 2021

Accepted: Jan 27, 2022

DOI: https://doi.org/10.2478/udt-2022-0003

Keywords
Diophantine approximation, oscillating sequences, irrationality measure, continued fractions, Ostrowski expansion

© 2022 Ioannis Tsokanos, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Density of Oscillating Sequences in the Real Line

Published Online: May 31, 2022

Page range: 105 - 130

Received: May 21, 2021

Accepted: Jan 27, 2022

DOI: https://doi.org/10.2478/udt-2022-0003

KeywordsDiophantine approximation, oscillating sequences, irrationality measure, continued fractions, Ostrowski expansion

© 2022 Ioannis Tsokanos, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Keywords
Diophantine approximation, oscillating sequences, irrationality measure, continued fractions, Ostrowski expansion